- Email: [email protected]
The Percus-Yevick-II equations for a binary mixture of fluids
Physica 39 17-20
Penn, S. 1968
THE
PERCUS-YEVICK-II
FOR A BINARY
EQUATIONS
MIXTURE
OF FLUIDS
*)
by S. PENN Laboratory
of Atomic and Solid State Physics, Cornell University, Ithaca, New York, U.S.A.
synopsis We present the Percus-Yevick-II integral equations for the pair correlation functions for a binary mixture of fluids. We also obtain certain simplified approximate forms of these equations by use of the superposition approximation.
1. Introdzcction. It has been recognized recently that the Percus-YevickII integral equationr) for the pair correlation function gives an accurate description of classical fluids at moderate densitiess). The Percus-Yevick-I (PY-I) equations for a binary mixture have been derived by Lebowit za). For several reasons, it seems worthwhile to derive the PY-II equations for a binary mixture. In particular, in addition to their use in connection with classical fluid mixtures, these equations can be employed to calculate the number of particles in the condensate in liquid 4He at T = O”K, and can also be applied to mixtures of aHe and 4He. In section 2, we recall the PY-II equation for the single-component fluid, and then outline the derivation of the equations for the binary mixture. In section 3, we briefly discuss certain simplifications of these equations which are achieved by use of the superposition approximation. We shall not include any details of the calculations; these may be found in the papers of Verlet and Lebowitz and the references contained therein. 2. Derivation. The PY-II equation for the pair correlation pure fluid is given by Verlet as a set of simultaneous integral g&O)
.@(l*O) = g(1, 0) -
k(l, 0) = C(l,O) @(l,O)
function in a equations:
C(1, 0) + @(l, O),
+ j d2 C(1, 2) ,412(2, O),
= [email protected](2,0)
C(3,O)
X [6(1> 4) -
-
17 -
(2)
x
PC(l, 411 a(% 3; 4) g(2, 31,
*) Work supported by the Advanced Research Projects Center at Cornell University, MSC Report #843.
(1)
(3)
Agency through the Materials Science
18
S. PENN
a(1,2;0)
=
g(l, 2, 0) g(l* 2)
-
1-
/G-(0,1) -
12(0, 2).
(4)
The symbols have the following meanings: g is the pair correlation function which we seek; h = g - 1; go is the interparticle potential (it is assumed that there are only pair interactions) ; p = l/kT; 0 is the origin of coordinates; 1, 2, . . . = rl, t-2, . . . . The function C(1, 0), introduced as a convenient intermediary in the equations, is the Ornstein-Zernike direct correlation function. The function a(1, 2; 0) is seen to contain the three-particle correlation function g(l,2, 0), which is of course unknown. Eqs. (l)-(4) are therefore not closed as they stand; in order to arrive at a closed set of equations for g( 1, 0), the function g( 1,2, 0) [or equivalently a( 1, 2; 0)] must be determined in some approximate manner. Verlet has presented two (approximate) integral equations for a( 1, 2; 0), the more successful*) of which is ~(1, 2; 0) = h(O, 1) h(O, 2) + g(0, 1) g(Q 2) j d4pC(Q
4) 4,
2; 4).
(5)
We shall show later how this equation is obtained. We now briefly outline the method used to obtain the analogous equations for a classical binary mixture. The procedure is that described by Lebowitz, simply carried to the next order. The system of interest is a classical fluid consisting of two kinds of particles, which we denote as type 1 and type 2. There are then three interparticle potentials qu(rfj), i, j = 1, 2 [vi, G VIZ], depending only on the distances ytj between respective particle pairs. Now consider this same system in the presence of two static external fields, each of which couples to only one type of particle. Denote the potentials of these external fields by ~1 and ~2, where q~ couples to type i particles. The properties of the fluid mixture are now all functionals of ~1 and q3s. (Functional dependence on ~1 and q~swill be indicated by a superscript (v).) For example, the density of each type of particle is such a functional: n?)(r), j = 1, 2. It is sometimes convenient to regard ~1 and ~2 as functionals of np) and n8). The Percus-Yevick equations are generated by performing a functional Taylor expansion of the functional n?)(l) eSpro), about the function values ~1 = ~2 = 0. Truncation after two terms gives the PY-I equations, and after three, the PY-II equations. After carrying out the expansion, the resulting expression is evaluated for the following choice of ‘pt: (J%(l) + %j(l, that is, the external potentials type j placed at the origin.
become
0);
the potentials
(6) due to a particle of
*) Success is judged entirely by the numerical results, as compared with monte car10 and molecular dynamics calculations. See ref. 2.
PERCUS-YEVICK-II
Using
the grand
EQUATION
canonical
FOR A BINARY
ensemble,
MIXTURE
it is easy to sees)
that
19 with this
choice, we have (7) It is also clear that the choice vi(l) --f IJQ(~, 0) brings us back again to a homogeneous, isotropic system*) for which gd~= gu(ru), nj(r) = p;r = constant, and so on. The results of the expansion, in the following manner:
truncated
gts( 1) 0) (pJJr.(l,O)= g&l, Aas(l> 0) = Czs(l,O) @zs(l, 0) = Q ;
Zkj= 1
x
[h
a(zS(l) 2; 0) =
Wl>
+
after three terms, may be written
0) ;
k=l
Ci,(l,
0) + @ts(l, O),
J d2 L(l,
2) p&&2,
j d2 d3 d4 pzCzs(2, 0) pzJrs(3, 4)
@z$2;)0) ,
-
PjCZj(l,
-
1-
0),
(8) (9)
0) x
4)lazkjp,3; 4)gzk(2,3), (10) h(O,
1) -
~8Z(O,2).
(11)
In order to close eqs. (8)-(1 l), we follow Verletl) and make a first order functional Taylor expansion of the functional g($(2,3). The result of this calculation is an integral equation for the function azki(2, 3; 1) : az&,
3; 1) = hdZ(l, 2) h&l, x
3) + &z(I, 2) &k(l, 3) x
i ,5 d4 C,( 1, 4) ~jazk;r(2>3 ; 4).
(14
Eqs. (8)-(11) are the exact results of the PY-II method. Combined with (12), these equations form a closed set for the calculation of giS. Eqs. (8)-(12) are very similar in structure to Verlet’s eqs. (l)-(4) and (9) 1). 3. Superposition. approximatiofz. Eqs. (8)-( 12) are suitable for the numerical calculation of gis as they stand. For a pure substance, they have yielded good results at low and moderate densitiess) 4). Recent works) has indicated that in treating problems which correspond to a fairly low density classical fluid, simplification of eqs. (l)-(5) can be obtained by use of the superposition approximation, without sacrifice of numerical success. These results encourage us to believe that the same approximation applied to the equations for a mixture will be equally successful. We first observe that eqs. (8)-( 11) can be closed directly, without the use *) This is of course not the case at intermediate stages of the calculation, when we have arbitrary values of (PI, ~2.
20
PERCUS-YEVICK-II
EQUATION
FOR A BINARY
MIXTURE
of (12), by using the superposition approximation in the form gzzcj(2,3, 4) = gzt(2, 3) gn(3, 4) grz(4, 2).
(13)
Substitution of (13) into (11) yields EZkj(2, 3; 4) -
&j(3, 4) JzZj(2,4),
(14)
S.A.
and hence
Q&(1,0) -+ 8X J d2 d3 d4 [&j 6(1, 4) - pjCzj(l, 4)] x S.A.
Zkj
x
PZCZ&
0)
PkCR8(3>0) gzzc(2,3) &(3,
4) hzj(2, 4).
(15)
Here d5,, is expressed as a triple integral. For purposes of ease of computation, it is preferable to have only a double integral. Such a reduction can be accomplished, as Verlet has shown, by combining the integral equation (12) with the superposition approximation (14). This manipulation gives ;
i=l
1 d4 p&(1 >4) az&
3; 4) --+ o. S.A.
Substitution of (14) and (16) into (IO), together with collapse of the term 6~ 8( 1, 4), gives finally the simple closure for @i8: %(l>O) =s~;sd2d3pzCz,(2,O)pkC~s(3,O)gzk(2,3)JZiz(l,2)Jtlk(l,3). Zk
(17)
We again notice that this approximate form for @is is structurally similar to the corresponding form given by V e r 1et 1). Acknowledgements. I would like to thank Professor G. V. Chester for suggesting this problem and for stimulating conversations, and also Dr. W. Francis for many helpful suggestions. Received 19-12-67
REFERENCES
1) Verlet, 4 Verlet,
L., Physica 31 (1965) 959. L. and Levesque, D., On the Theory of Classical Fluids - VI (preprint). J. L., Phys. Rev. 133 (1964) A595 3) Lebowitz, D., Schiff, D., and Verlet, L., On the Ground State of Liquid and 4) Khiet, T., Levesque, Solid Helium Four at T = 0°K (preprint). 5) Chester, G. V., Francis, W. and Reatto, L., The Ground State Properties of Liquid Helium Four, to be published.
Penn, S. 1968
THE
PERCUS-YEVICK-II
FOR A BINARY
EQUATIONS
MIXTURE
OF FLUIDS
*)
by S. PENN Laboratory
of Atomic and Solid State Physics, Cornell University, Ithaca, New York, U.S.A.
synopsis We present the Percus-Yevick-II integral equations for the pair correlation functions for a binary mixture of fluids. We also obtain certain simplified approximate forms of these equations by use of the superposition approximation.
1. Introdzcction. It has been recognized recently that the Percus-YevickII integral equationr) for the pair correlation function gives an accurate description of classical fluids at moderate densitiess). The Percus-Yevick-I (PY-I) equations for a binary mixture have been derived by Lebowit za). For several reasons, it seems worthwhile to derive the PY-II equations for a binary mixture. In particular, in addition to their use in connection with classical fluid mixtures, these equations can be employed to calculate the number of particles in the condensate in liquid 4He at T = O”K, and can also be applied to mixtures of aHe and 4He. In section 2, we recall the PY-II equation for the single-component fluid, and then outline the derivation of the equations for the binary mixture. In section 3, we briefly discuss certain simplifications of these equations which are achieved by use of the superposition approximation. We shall not include any details of the calculations; these may be found in the papers of Verlet and Lebowitz and the references contained therein. 2. Derivation. The PY-II equation for the pair correlation pure fluid is given by Verlet as a set of simultaneous integral g&O)
.@(l*O) = g(1, 0) -
k(l, 0) = C(l,O) @(l,O)
function in a equations:
C(1, 0) + @(l, O),
+ j d2 C(1, 2) ,412(2, O),
= [email protected](2,0)
C(3,O)
X [6(1> 4) -
-
17 -
(2)
x
PC(l, 411 a(% 3; 4) g(2, 31,
*) Work supported by the Advanced Research Projects Center at Cornell University, MSC Report #843.
(1)
(3)
Agency through the Materials Science
18
S. PENN
a(1,2;0)
=
g(l, 2, 0) g(l* 2)
-
1-
/G-(0,1) -
12(0, 2).
(4)
The symbols have the following meanings: g is the pair correlation function which we seek; h = g - 1; go is the interparticle potential (it is assumed that there are only pair interactions) ; p = l/kT; 0 is the origin of coordinates; 1, 2, . . . = rl, t-2, . . . . The function C(1, 0), introduced as a convenient intermediary in the equations, is the Ornstein-Zernike direct correlation function. The function a(1, 2; 0) is seen to contain the three-particle correlation function g(l,2, 0), which is of course unknown. Eqs. (l)-(4) are therefore not closed as they stand; in order to arrive at a closed set of equations for g( 1, 0), the function g( 1,2, 0) [or equivalently a( 1, 2; 0)] must be determined in some approximate manner. Verlet has presented two (approximate) integral equations for a( 1, 2; 0), the more successful*) of which is ~(1, 2; 0) = h(O, 1) h(O, 2) + g(0, 1) g(Q 2) j d4pC(Q
4) 4,
2; 4).
(5)
We shall show later how this equation is obtained. We now briefly outline the method used to obtain the analogous equations for a classical binary mixture. The procedure is that described by Lebowitz, simply carried to the next order. The system of interest is a classical fluid consisting of two kinds of particles, which we denote as type 1 and type 2. There are then three interparticle potentials qu(rfj), i, j = 1, 2 [vi, G VIZ], depending only on the distances ytj between respective particle pairs. Now consider this same system in the presence of two static external fields, each of which couples to only one type of particle. Denote the potentials of these external fields by ~1 and ~2, where q~ couples to type i particles. The properties of the fluid mixture are now all functionals of ~1 and q3s. (Functional dependence on ~1 and q~swill be indicated by a superscript (v).) For example, the density of each type of particle is such a functional: n?)(r), j = 1, 2. It is sometimes convenient to regard ~1 and ~2 as functionals of np) and n8). The Percus-Yevick equations are generated by performing a functional Taylor expansion of the functional n?)(l) eSpro), about the function values ~1 = ~2 = 0. Truncation after two terms gives the PY-I equations, and after three, the PY-II equations. After carrying out the expansion, the resulting expression is evaluated for the following choice of ‘pt: (J%(l) + %j(l, that is, the external potentials type j placed at the origin.
become
0);
the potentials
(6) due to a particle of
*) Success is judged entirely by the numerical results, as compared with monte car10 and molecular dynamics calculations. See ref. 2.
PERCUS-YEVICK-II
Using
the grand
EQUATION
canonical
FOR A BINARY
ensemble,
MIXTURE
it is easy to sees)
that
19 with this
choice, we have (7) It is also clear that the choice vi(l) --f IJQ(~, 0) brings us back again to a homogeneous, isotropic system*) for which gd~= gu(ru), nj(r) = p;r = constant, and so on. The results of the expansion, in the following manner:
truncated
gts( 1) 0) (pJJr.(l,O)= g&l, Aas(l> 0) = Czs(l,O) @zs(l, 0) = Q ;
Zkj= 1
x
[h
a(zS(l) 2; 0) =
Wl>
+
after three terms, may be written
0) ;
k=l
Ci,(l,
0) + @ts(l, O),
J d2 L(l,
2) p&&2,
j d2 d3 d4 pzCzs(2, 0) pzJrs(3, 4)
@z$2;)0) ,
-
PjCZj(l,
-
1-
0),
(8) (9)
0) x
4)lazkjp,3; 4)gzk(2,3), (10) h(O,
1) -
~8Z(O,2).
(11)
In order to close eqs. (8)-(1 l), we follow Verletl) and make a first order functional Taylor expansion of the functional g($(2,3). The result of this calculation is an integral equation for the function azki(2, 3; 1) : az&,
3; 1) = hdZ(l, 2) h&l, x
3) + &z(I, 2) &k(l, 3) x
i ,5 d4 C,( 1, 4) ~jazk;r(2>3 ; 4).
(14
Eqs. (8)-(11) are the exact results of the PY-II method. Combined with (12), these equations form a closed set for the calculation of giS. Eqs. (8)-(12) are very similar in structure to Verlet’s eqs. (l)-(4) and (9) 1). 3. Superposition. approximatiofz. Eqs. (8)-( 12) are suitable for the numerical calculation of gis as they stand. For a pure substance, they have yielded good results at low and moderate densitiess) 4). Recent works) has indicated that in treating problems which correspond to a fairly low density classical fluid, simplification of eqs. (l)-(5) can be obtained by use of the superposition approximation, without sacrifice of numerical success. These results encourage us to believe that the same approximation applied to the equations for a mixture will be equally successful. We first observe that eqs. (8)-( 11) can be closed directly, without the use *) This is of course not the case at intermediate stages of the calculation, when we have arbitrary values of (PI, ~2.
20
PERCUS-YEVICK-II
EQUATION
FOR A BINARY
MIXTURE
of (12), by using the superposition approximation in the form gzzcj(2,3, 4) = gzt(2, 3) gn(3, 4) grz(4, 2).
(13)
Substitution of (13) into (11) yields EZkj(2, 3; 4) -
&j(3, 4) JzZj(2,4),
(14)
S.A.
and hence
Q&(1,0) -+ 8X J d2 d3 d4 [&j 6(1, 4) - pjCzj(l, 4)] x S.A.
Zkj
x
PZCZ&
0)
PkCR8(3>0) gzzc(2,3) &(3,
4) hzj(2, 4).
(15)
Here d5,, is expressed as a triple integral. For purposes of ease of computation, it is preferable to have only a double integral. Such a reduction can be accomplished, as Verlet has shown, by combining the integral equation (12) with the superposition approximation (14). This manipulation gives ;
i=l
1 d4 p&(1 >4) az&
3; 4) --+ o. S.A.
Substitution of (14) and (16) into (IO), together with collapse of the term 6~ 8( 1, 4), gives finally the simple closure for @i8: %(l>O) =s~;sd2d3pzCz,(2,O)pkC~s(3,O)gzk(2,3)JZiz(l,2)Jtlk(l,3). Zk
(17)
We again notice that this approximate form for @is is structurally similar to the corresponding form given by V e r 1et 1). Acknowledgements. I would like to thank Professor G. V. Chester for suggesting this problem and for stimulating conversations, and also Dr. W. Francis for many helpful suggestions. Received 19-12-67
REFERENCES
1) Verlet, 4 Verlet,
L., Physica 31 (1965) 959. L. and Levesque, D., On the Theory of Classical Fluids - VI (preprint). J. L., Phys. Rev. 133 (1964) A595 3) Lebowitz, D., Schiff, D., and Verlet, L., On the Ground State of Liquid and 4) Khiet, T., Levesque, Solid Helium Four at T = 0°K (preprint). 5) Chester, G. V., Francis, W. and Reatto, L., The Ground State Properties of Liquid Helium Four, to be published.